# Rigorous definition and relations between point/vector/affine space/vector space/basis/frame/coordinate system

I am trying to understand the exact relation between all these things:

Can you explain me rigorously (in the mathematical sense) all the inter-relations existing between these geometrical concepts?

Additionnally, a strange thing is that this french wikipedia article seems to define a frame as a basis + origin whereas this english wikipedia article seems to define the relation between a frame and a basis completely differently. So could you explain me the right relation between a basis and a frame (and why one of the article may be erroneous).

• Which of this concepts you have trouble with? Do you know the elementary set theory? Linear algebra? If not, pick up any book on linear algebra and start reading. Commented Feb 27, 2014 at 7:01
• I have good notions of set theory and linear algebra and I do not have big problem on these concepts taken separetely. But I do not have a clear picture of how to assemble and construct a coherent landscape with these concepts. Commented Feb 27, 2014 at 7:05
• @Vincent I highly recommend you take a look at Linear Algebra and Geometry by Igor R. Shafarevich & Alexey O. Rezimov. It explains and describes the relationships between all these structures very lucidly. Commented Oct 16, 2014 at 8:23